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The goal of this work is to study the cool, active binary star $\sigma^2$CrB, focussing on its magnetic field. We used Stokes $IV$ data from the twinspectropolarimeters Narval at the TBL and ESPaDOnS at the CFHT. Theleast-squares deconvolution multi-line technique was used to increase thesignal-to-noise ratio of the data. We then applied a new binary Zeeman-Dopplerimaging code to reconstruct simultaneously the magnetic topology and brightnessdistribution of both components. This analysis was carried out for twoobservational epochs in 2014 and 2017. A previously unconfirmed magnetic fieldof the primary star has been securely detected. The polarisation signatures ofthe secondary appear to have a systematically larger amplitude than that of theprimary. This corresponds to a stronger magnetic field, for which the magneticenergy of the secondary exceeds that of the primary by a factor of 3.3-5.7.While the magnetic energy is similar for the secondary star in the two epochs,the magnetic energy is about twice as high in 2017 for the primary. Themagnetic field topology of the two stars in the earlier epoch (2014) is verydifferent. In the earlier epoch, the magnetic field at the visible pole appearsto be of opposite polarity for the primary and secondary, suggesting linkedmagnetospheres. The apparent rotational periods of both $\sigma^2$ CrBcomponents are longer than the orbital period, which we interpret as anevidence of a solar-like differential rotation. Despite their nearly identicalfundamental parameters, the components of $\sigma^2$ CrB system exhibitdifferent magnetic field properties. This indicates that the magnetic dynamoprocess is a very sensitive function of stellar parameters.

Magnetic field topology of the cool, active, short-period binary system $\sigma^2$ Coronae Borealis